document.write( "Question 559973: a triangle is an isosceles triangle but not a right triangle. For some value x the three sides are equal to sin(x), cos(x), and tan(x). find the value of sin(x). \n" ); document.write( "
Algebra.Com's Answer #363609 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Since the sides are sin(x), cos(x), and tan(x), and\r\n" );
document.write( "the triangle is isosceles, two of the sides are equal, \r\n" );
document.write( "but we don't know which two.  So we try all three \r\n" );
document.write( "possibilities:\r\n" );
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document.write( "    1.  We try sin(x) = cos(x) \r\n" );
document.write( "Divide both sides by cos(x)\r\n" );
document.write( "               \"sin%28x%29%2Fcos%28x%29\" = 1\r\n" );
document.write( "                    x = 45° or \"pi%2F4\"\r\n" );
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document.write( "so the sides would be \r\n" );
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document.write( "               sin(\"pi%2F4\") = \"sqrt%282%29%2F2\"\r\n" );
document.write( "               cos(\"pi%2F4\") = \"sqrt%282%29%2F2\"\r\n" );
document.write( "               tan(\"pi%2F4\") = 1\r\n" );
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document.write( "But that is a right triangle, and don't want that answer.\r\n" );
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document.write( "    2.  We try sin(x) = tan(x) \r\n" );
document.write( "That can only have solution x=0,\"pi\"\r\n" );
document.write( "But those cannot be angles of a triangle.\r\n" );
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document.write( "    3.  We try cos(x) = tan(x)\r\n" );
document.write( "               cos(x) = \"sin%28x%29%2Fcos%28x%29\"\r\n" );
document.write( "Multiply through by cos(x)\r\n" );
document.write( "              cosē(x) = sinē(x)\r\n" );
document.write( "          1 - sinē(x) = sinē(x)\r\n" );
document.write( "-sinē(x) - sin(x) + 1 = 0\r\n" );
document.write( "Multiply through by -1\r\n" );
document.write( " sinē(x) + sin(x) - 1 = 0   \r\n" );
document.write( "  \r\n" );
document.write( "               sin(x) = \"%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\" \r\n" );
document.write( "               sin(x) = \"%28-1+%2B-+sqrt%28+1%5E2-4%2A1%2A%28-1%29+%29%29%2F%282%2A1%29+\"\r\n" );
document.write( "               sin(x) = \"%28-1+%2B-+sqrt%28+1%2B4+%29%29%2F2+\"\r\n" );
document.write( "               sin(x) = \"%28-1+%2B-+sqrt%285%29%29%2F2+\"\r\n" );
document.write( "sin(x) must be positive, so           \r\n" );
document.write( "               sin(x) = \"%28-1+%2B+sqrt%285%29%29%2F2+\" (a.k.a \"the golden ratio\")\r\n" );
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document.write( "Drawn to scale here is that triangle:\r\n" );
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document.write( "Figures involving the golden ratio are supposed to be \"the most\r\n" );
document.write( "beautiful\".  So maybe this triangle is \"the most beautiful one.\"  \r\n" );
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document.write( ":-)\r\n" );
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document.write( "Edwin
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